for which Hamilton's equations of motion simplify in the new phase space (\(\mathbf {\alpha }, J\) ). By restricting ourselves to a canonical transformation, we preserve the form of Vlasov's equation. The coordinates \(\mathbf {\alpha }\)  and \(J\)  respectively correspond to the action angles and adiabatic invariants of our system. It is well known [1]} that Hamilton's equations of motion then reduce to
\(\frac{\partial H}{\partial \mathbf {\alpha }} &= - \dot{J} = 0,\\\frac{\partial H}{\partial J} &= \dot{\mathbf {\alpha }} = \Omega ,\) 
